Normal maps - (non)-examples (Ex)

Latest revision as of 07:43, 10 January 2019

1a) Give an example of a degree one map of closed $<wikitex>; 1a) Give an example of a degree one map of closed $n$-manifolds $f \colon M \to X$ which cannot be covered by a map $\overline{f} \colon \nu_M \to \nu_X$ of normal bundles. 1b) Give an example of a degree one map of closed $n$-manifolds $f \colon M \to X$ which cannot be covered by a map $\overline{f} \colon \nu_M \to \xi$ of bundles, for any stable bundles $\xi$. 2) For every integer $d$, give an example of a degree $d$ map $f_d \colon M \to X$ of closed $n$-manifolds which can be covered by a map $\overline{f_d} \colon \nu_M \to \nu_X$ of normal bundles. 3) Let $F_g$ denote the oriented surface of genus $g$. Determine the values of $(g, g')$ for which there is a degree one normal map $(f, \overline{f}) \colon F_g \to F_{g'}$. </wikitex> <!-- == References== {{#RefList:}} --> [[Category:Exercises]] [[Category:Exercises without solution]]n$-manifolds $f \colon M \to X$ which cannot be covered by a map $\overline{f} \colon \nu_M \to \nu_X$ of normal bundles.

1b) Give an example of a degree one map of closed $n$-manifolds $f \colon M \to X$ which cannot be covered by a map $\overline{f} \colon \nu_M \to \xi$ of bundles, for any stable bundle $\xi$.

2) For every integer $d$, give an example of a degree $d$ map $f_d \colon M \to X$ of closed $n$-manifolds which can be covered by a map $\overline{f_d} \colon \nu_M \to \nu_X$ of normal bundles.

3) Let $F_g$ denote the oriented surface of genus $g$. Determine the values of $(g, g')$ for which there is a degree one normal map $(f, \overline{f}) \colon F_g \to F_{g'}$.